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CHAPTER
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5 EDGE DETECTION
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(a) Figure 523: The result of edge detection applied to the test image from Figure 522 with a 7 x 7 Gaussian smoothing filter The picture in part (a) is a gray level image of the result after smoothing with a Gaussian smoothing filter, computing the gradient approximation, and suppressing nonmaxima The weak edge fragments due to noise and fine texture do not show up clearly Part (b) is a plot of the image from part (a) with every pixel that was greater than zero drawn in black This plot shows the weak edge fragments that are present in the result of edge detection but do not show up clearly in a gray level display of the results the step edge is not ideal but makes a gradual transition from one level to another, then the result of Gaussian smoothing and a first derivative can be approximated by a broader Gaussian Consider a set of measurements drawn from a normal distribution The center of the bell-shaped curve corresponds to the mean value of the normal distribution, which can be estimated by averaging the measurements Now suppose that a histogram of the measurements, rather than the raw measurements themselves, is all the information that is available The mean can be estimated by dividing the sum of the values for the centers of the histogram buckets, weighted by the number of entries in each bucket, by the area of the histogram By analogy, the location of an edge can be estimated to subpixel resolution by averaging along the profile in the output of the Gaussian edge
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57 SUBPIXEL LOCATION ESTIMATION
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Figure 524: The result of edge detection applied to the test image from Figure 522 A 31 x 31 Gaussian filter was used to smooth the image, followed by computation of the gradient approximation and then nonmaxima suppression Part (a) displays the nonmaxima-suppressed result as a gray level image Note how the large smoothing filter has rounded the edges Part (b) plots the results to show all of the edge fragments
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detector To compute the edge location to subpixel resolution, take samples of the magnitude of the Gaussian edge detector output (without nonmaxima suppression) along"the gradient direction to either side of the edge until the gradient magnitude falls below a threshold Use the samples of the gradient magnitude 9i as weights to compute the weighted sum of the position di along the gradient The subpixel correction to the position of the edge along the gradient direction is given by (550)
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where di is the distance of a pixel along the gradient from the pixel where the edge was detected, and 9i is the gradient magnitude
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CHAPTER 5 EDGE DETECTION
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A simpler algorithm that is probably as effective is to compute the position (bX,by) of the edge to subpixel resolution, relative to the pixel where the edge was detected, by applying a first moment calculation to the magnitude of the gradient of the Gaussian edge detector The correction can be added to the coordinates of the pixel to provide a more accurate estimate of edge location The algorithm for computing the average along the profile of gradient magnitude, though more complex, has the advantage that the profile can be compared with the ideal profile using statistical techniques and the results of this comparison can be used as the criterion for edge detection If the profile is not close to a Gaussian, the edge does not correspond to the ideal step model It may not be possible in this case to accurately estimate the edge location using the techniques presented in this section In 12, we will present methods for calibrating the coordinate system of the image The integer coordinates of a pixel [i,j] can be mapped to coordinates (x, y) in the image plane of the camera The correction to the edge location provided by either of the methods for estimating edge location to subpixel resolution described above is added to the (x, y) coordinates of the pixel to obtain the precise location of the edge in the image plane This precise, calibrated value is what is needed for measuring feature dimensions
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